# Statistical limit laws for hyperbolic groups

## TL;DR

This paper develops statistical limit laws for functions on hyperbolic groups using ergodic theory, including central and local limit theorems, with applications to group actions on negatively curved spaces.

## Contribution

It introduces new limit theorems for hyperbolic groups, extending to multidimensional cases and providing precise statistical comparisons for group actions.

## Key findings

- Central limit theorem for abelianisation map
- Local limit theorems for group homomorphisms
- Statistical comparison between word length and displacement

## Abstract

Using techniques from ergodic theory and symbolic dynamics, we derive statistical limit laws for real valued functions on hyperbolic groups. In particular, our results apply to convex cocompact group actions on $\text{CAT}(-1)$ spaces, and provide a precise statistical comparison between word length and displacement. After generalising our methods to the multidimensional setting, we prove that the abelianisation map satisfies a non-degenerate multidimensional central limit theorem. We also obtain local limit theorems for group homomorphisms and for the displacement function associated to certain actions.